A multidisciplinary coupling relationship coordination algorithm using the hierarchical control methods of complex systems and its application in multidisciplinary design optimization

Abstract

Because of the increasing complexity in engineering systems, multidisciplinary design optimization has attracted increasing attention. High computational expense and organizational complexity are two main challenges of multidisciplinary design optimization. To address these challenges, the hierarchical control method of complex systems is developed in this study. Hierarchical control method is a powerful way which has been utilized widely in the control and coordination of large-scale complex systems. Here, a hierarchical control method–based coupling relationship coordination algorithm is proposed to solve multidisciplinary design optimization problems. Coupling relationship coordination algorithm decouples the involved disciplines of a complex system and then optimizes each discipline objective at sub-system level. Coupling relationship coordination algorithm can maintain the consistency of interaction information (or in other words, sharing design variables and coupling design variables) in different disciplines by introducing control parameters. The control parameters are assigned by the coordinator at system level. A mechanical structure multidisciplinary design optimization problem is solved to illustrate the details of the proposed approach.

Introduction

Modern design methods for complex engineering systems have obtained increasing attention to achieve relatively better design solutions recently.^1–7 There are usually many different sub-systems (or in other words, disciplines) in a complex system. It is critical to transform an engineering optimization problem into its corresponding mathematical model. If the formulation in mathematical model is improper, it may result in unacceptable design solutions. Traditional design optimization methods can solve the corresponding design problems of sub-systems individually. However, it is generally difficult for the traditional design optimization methods to find the optimal solutions of whole system.^8–10 Thus, more advantageous design strategies are necessary and should be developed.

Multidisciplinary design optimization (MDO) is powerful to satisfy the above requirements. The definition of MDO can be depicted as one strategy for the arrangement of large-scale organization in which the high dimension of interactive information between coupling components motivates designers to manipulate input information in different sub-systems simultaneously.^11–13 In MDO, a system optimization problem can be divided into some small-scale sub-problems design problems. A coordination strategy is utilized to drive all design variables in sub-problems toward the solutions which are also optimal for original system optimization problem. It will be more convenient to organize different small-scale discipline models and provide necessary communication if a complex system can be divided into many sub-systems. Hence, decomposing large-scale systems facilitates the concurrent design in modern engineering.^14,15

Many hierarchical and nonhierarchical manners have been proposed to address issues of MDO. For nonhierarchical manners, such as multiple-discipline-feasible method, necessary iterations are conducted to satisfy that multidisciplinary analysis (MDA) problems can be solved successfully. ¹⁶ In the all-at-once method, feasibility can be guaranteed if optimization convergence is obtained. ¹⁷ The individual discipline feasible approach can keep the feasibility of individual discipline analysis during the whole optimization process. Furthermore, the multidisciplinary feasibility (MDF) of complex systems can be satisfied when the convergence of MDO problems is obtained. ¹⁸ The concurrent subspace optimization (CSSO) method uses global sensitivity equations to assign different disciplinary design optimization problems to the corresponding professional design team. Global coordination modules are required in CSSO to keep the MDF of MDO. ¹⁹ For hierarchical manner, bi-level integrated system synthesis connects system and sub-system objectives using global sensitivity equation (GSE). Thus, MDO solutions can be obtained by solving discipline design optimization problems. ²⁰ In the analytical target cascading (ATC) method, all target value of top level variables are propagated to appropriate format for different disciplines in a consistent way. ²¹ The collaborative optimization (CO) method minimizes the deviations between target value from system level and response value from sub-system level using compatibility constraints.^22–29

In this study, a coupling relationship coordination algorithm (CRCA) is integrated into the framework of MDO based on large-scale systems control and coordination strategy. In CRCA, whole system is divided into several sub-systems (disciplines) along discipline boundaries. The system-level coordinator provides initial values of control parameters as the optimization parameters of discipline optimization problems. Then, the solutions of different discipline optimization problems can be obtained and provided to system-level coordinator for updating control parameters. After that, all updated control parameters are provided to each discipline optimization problems again to minimize differences between interaction variables from different disciplines. Compared with the existing MDO methods, CRCA allows that the system coordinator can just focus on updating the control parameters to maintain the interdisciplinary consistency at system level. This process treats the design variables as design parameters and does not be involved in the objective optimization directly. At the same time, discipline optimizers can focus on optimizing the sub-system objectives in each discipline independently. During CRCA process, all control parameters are treated as design parameters and do not affect the other disciplines’ optimization processes mutually.

The rest of this study is organized as follows. The MDO problem is introduced in section “The MDO problem.” Meanwhile, MDF and CO are also briefly reviewed as comparison methods. The details of proposed approach are given in section “CRCA for MDO problem,” including the formulation and the procedure. In section “Test example,” a design problem of speed reducer is utilized to illustrate the application of proposed method. Conclusions are given in section “Conclusion.”

The MDO problem

In a complex system, there is more than one sub-system (discipline). The outputs of a discipline may be the inputs of other disciplines. As shown in Figure 1, in a multidisciplinary system, there are many coupling sub-systems. These coupling sub-systems can be depicted by discipline analysis. An interacting sub-system not only has a certain degree of disciplinary autonomy but also depends on other sub-systems because of sharing variables and coupling variables.

Figure 1.

MDO discipline decomposition.

Correspondingly, in this study, the formulation of an MDO problem is given as

$$\begin{array}{c}\underset{{\mathbf{X}}_s,{\mathbf{X}}_i,{\mathbf{Y}}_{•i},{\mathbf{Y}}_{i•}}{min}f=\sum _{i=1}^n{f}_i({\mathbf{X}}_i,{\mathbf{X}}_s,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})\hfill \\ \mathrm{s}.\mathrm{t}.{\mathit{h}}_i({\mathbf{X}}_i,{\mathbf{X}}_s,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})=0\hfill \\ {\mathit{g}}_i({\mathbf{X}}_i,{\mathbf{X}}_s,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})\ge 0\hfill \\ {\mathbf{Y}}_{i•}={\mathbf{Y}}_{i•}({\mathbf{X}}_i,{\mathbf{X}}_s,{\mathbf{Y}}_{•i})\hfill \\ \begin{array}{cc}{\mathbf{X}}_i^{\min }\le {\mathbf{X}}_i\le {\mathbf{X}}_i^{\max }\hfill & {\mathbf{X}}_s^{\min }\le {\mathbf{X}}_s\le {\mathbf{X}}_s^{\max }\hfill \\ \multicolumn{1}{c}{{\mathbf{Y}}_{i•}^{\min }\le {\mathbf{Y}}_{i•}\le {\mathbf{Y}}_{i•}^{\max }}& {\mathbf{Y}}_{•i}^{\min }\le {\mathbf{Y}}_{•i}\le {\mathbf{Y}}_{•i}^{\max }\hfill \\ \multicolumn{1}{c}{i=1,2,\dots ,n}\end{array}\hfill \end{array}$$

(1)

where $f(•)$ is system objective; $f_i(•)$ is discipline objective; ${\mathit{h}}_i(•)$ and ${\mathit{g}}_i(•)$ are vectors of equality and inequality constraints in the ith discipline, respectively; ${\mathbf{X}}_i$ is a set of discipline variables of the ith discipline; ${\mathbf{X}}_s$ is a set of sharing variables; $\mathbf{Y}$ is a set of coupling variables; $i•$ and $•i$ mean the outputs and inputs of the ith discipline, respectively; n is the number of disciplines; and ${\mathbf{Y}}_{i•}={\mathbf{Y}}_{i•}({\mathbf{X}}_i,{\mathbf{X}}_s,{\mathbf{Y}}_{•i})$ denotes the discipline analysis. In this study, we suppose that the system objective is the linear sum of all discipline objectives.

The review of MDF

MDF is a classic MDO approach, which is also one of the variable reduction methods. When using MDF to solve the optimization problem in equation (1), a nonlinear programming is required and introduced. The analysis for multidisciplinary system is considered as a black box. Meanwhile, all design information is treated as independent input. Thus, there is no need to change the formulation of equation (1). In this way, MDO problems can be considered as nonlinear design problems. ¹⁸ In each iteration of optimization, as shown in Figure 2, design variables are chosen as inputs of MDA. MDA usually represents the state equations ${\mathbf{Y}}_{i•}=f_i({\mathbf{X}}_s,{\mathbf{X}}_i,{\mathbf{Y}}_{•i})$ for interdisciplinary coupling and discipline analysis.

Figure 2.

The structure of MDF strategy.

MDF could be an efficient method to obtain accurate results when solving a simple problem. In each iteration of optimization, MDF can be maintained by MDA. However, it is usually difficult or impractical to utilize MDF because of the high complexity of engineering systems. First, the difficulty of analysis integration may result in heavy calculation burden for MDF. Second, MDA should be performed by computing derivatives in each iteration, if finite-difference derivatives are introduced. Moreover, the MDF algorithm is sensitive to the fixed-point iteration and the convergence of MDA, which may be detrimental to the robustness of MDF. ³⁰

The review of CO

There is a two-level structure in CO method. Compatibility constraints J exist at the system level and sub-system level, respectively. The compatibility constraints are used to coordinate the balance of coupling information between different disciplines. The responsibilities of analysis and design are assigned to every discipline. As shown in Figure 3, all disciplines are communicated by a system-level coordination routine. Specific discipline information does not have to be communicated with other disciplines. To assure that all discipline constraints can be satisfied in each local group, discipline information is allowed to disagree with other disciplines during discipline design process. All discipline objectives are to minimize the interdisciplinary discrepancies. The coordination optimizer at system level guarantees the discrepancies can vanish. More specifically, target values of $\mathbf{X}\mathbf{\prime }$ , ${\mathbf{X}\mathbf{\prime }}_s$ , and $\mathbf{Y}\mathbf{\prime }$ are provided by the coordination optimizer. However, the discipline responsibilities at sub-system level are to match these target values to the extent permitted by discipline constraints.

Figure 3.

The structure of CO strategy.

There are many advantages of CO. ³¹ In CO, compatibility constraints are utilized instead of MDA to satisfy the requirement of MDF at system level. Consequently, the discipline autonomy regarding optimization and analysis can be enjoyed in CO. The process of discipline analysis can also be conducted in parallel. Furthermore, CO can take advantage of different discipline analysis software to obtain MDF during the whole optimization process. However, it will be hard for CO to produce Karush-Kuhn-Tucker (KKT) points if very high nonlinear constraints are introduced. It is because that the number of compatibility constraints in CO is equal to coupling dimensions. Consequently, it will be suitable for CO to solve the MDO problems which have small number of interaction variables. Furthermore, the objective and constraints of original optimization problem must be assigned into system level and sub-system level, respectively. Thus, although every discipline optimization process can enjoy autonomy, explicit discipline objectives do not exist in each discipline optimization problem. Meanwhile, there are also no general constraints at system level.

CRCA for MDO problem

The original purpose of CRCA is to present the theory of control and coordination in large-scale complex systems. Its convergence condition is also proved by Findeisen et al. ³² Here, CRCA is introduced to solve MDO problems. In MDO problems, as shown in Figure 1, in order to realize the decoupling among different disciplines, a large-scale engineering system can be divided into some disciplines according to discipline boundaries. In this study, the ith discipline analysis is illustrated in Figure 4.

Figure 4.

The ith discipline analysis.

In Figure 4, the discipline coupling consistency requirements can be considered as coupling equations

$${\mathbf{Y}}_{i•}={\mathbf{Y}}_{i•}({\mathbf{D}}_i,{\mathbf{Y}}_{•i})\mathrm{and}{\mathbf{Y}}_{•i}={\mathbf{Y}}_{•i}\left({\mathbf{Y}}_{i•}\right)$$

(2)

It should be noted that, for discipline i, the coupling input ${\mathbf{Y}}_{•i}$ is just the function of coupling output ${\mathbf{Y}}_{i•}$ . For the easier application of CRCA, the distinction between sharing design variables ${\mathbf{X}}_s$ and discipline design variables ${\mathbf{X}}_i$ is not considered. Thus, both ${\mathbf{X}}_s$ and ${\mathbf{X}}_i$ can be treated as discipline input design variables ${\mathbf{D}}_i$ in discipline i. To simplify the notation, equation (1) can be rewritten as

$$\begin{array}{c}\underset{{\mathbf{D}}_i,{\mathbf{Y}}_{•i},{\mathbf{Y}}_{i•}}{min}f=\sum _{i=1}^n{f}_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})\hfill \\ \mathrm{s}.\mathrm{t}.{\mathbf{h}}_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})=0\hfill \\ {\mathbf{g}}_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})\ge 0\hfill \\ {\mathbf{Y}}_{i•}={\mathbf{Y}}_{i•}({\mathbf{D}}_i,{\mathbf{Y}}_{•i})\hfill \\ \begin{array}{cc}{\mathbf{D}}_i^{\min }\le {\mathbf{D}}_i\le {\mathbf{D}}_i^{\max }\hfill & {\mathbf{D}}_i=\{{\mathbf{X}}_s,{\mathbf{X}}_i\}\hfill \\ \multicolumn{1}{c}{{\mathbf{Y}}_{•i}^{\min }\le {\mathbf{Y}}_{•i}\le {\mathbf{Y}}_{•i}^{\max }}& {\mathbf{Y}}_{i•}^{\min }\le {\mathbf{Y}}_{i•}\le {\mathbf{Y}}_{i•}^{\max }\hfill \\ \multicolumn{1}{c}{i=1,2,\dots ,n}\end{array}\hfill \end{array}$$

(3)

Herein, we introduce control parameters $\mathbf{\lambda }$ . The Lagrange formulation of the system objective function in equation (3) can be stated as

$$L\left(D_i,Y_{i•},Y_{•i}\right)={\displaystyle \sum _{i=1}^n\left\{f_i\left(D_i,Y_{i•},Y_{•i}\right)+{\lambda }_i^T\times \left(Y_{•i}-Y_{•i}\left(Y_{i•}\right)\right)\right\}}$$

(4)

Obviously, the Lagrange formulation of system objective function could be equal to system objective function when all coupling equations are satisfied. Based on the discipline decomposition, equation (4) can be decomposed into several sub-Lagrange functions for different disciplines

$$L({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})=\sum _{i=1}^n{L}_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})$$

(5)

$$L_i\left(D_i,Y_{i•},Y_{•i}\right)=f_i\left(D_i,Y_{i•},Y_{•i}\right)+{\lambda }_i^T\times Y_{•i}-{\displaystyle \sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n{\lambda }_j^T\times Y_{i•}\left(D_i,Y_{•i}\right)}$$

(6)

In equations (5) and (6), the ith Lagrange formulation is the function of discipline inputs ${\mathbf{D}}_i$ , discipline coupling outputs ${\mathbf{Y}}_{i•}$ , and discipline coupling inputs ${\mathbf{Y}}_{•i}$ . After control parameters are assigned to disciplines for coordination, the ith discipline optimization problem can be solved by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Give}\mathbf{\lambda }=({\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2,\dots ,{\mathbf{\lambda }}_n)}\\ \multicolumn{1}{c}{\underset{{\mathbf{D}}_i,{\mathbf{Y}}_{•i},{\mathbf{Y}}_{i•}}{min}{L}_i=f_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})+{\mathbf{\lambda }}_i^T}\\ \times {\mathbf{Y}}_{•i}-\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^n{\mathbf{\lambda }}_j^T\times {\mathbf{Y}}_{i•}({\mathbf{D}}_i,{\mathbf{Y}}_{•i})\hfill \\ \multicolumn{1}{c}{\mathrm{s}.\mathrm{t}.h_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})=0}\\ g_i({\mathbf{D}}_i,{\mathbf{Y}}_{i•},{\mathbf{Y}}_{•i})\ge 0\hfill \\ {\mathbf{Y}}_{i•}={\mathbf{Y}}_{i•}({\mathbf{D}}_i,{\mathbf{Y}}_{•i})\hfill \\ {\mathbf{D}}^{\min }\le {\mathbf{D}}_i\le {\mathbf{D}}^{\max }{\mathbf{D}}_i=\{{\mathbf{X}}_s,{\mathbf{X}}_i\}\hfill \\ {\mathbf{Y}}_{•i}^{\min }\le {\mathbf{Y}}_{•i}\le {\mathbf{Y}}_{•i}^{\max }{\mathbf{Y}}_{i•}^{\min }\le {\mathbf{Y}}_{i•}\le {\mathbf{Y}}_{i•}^{\max }\hfill \\ i=1,2,\dots ,n\hfill \end{array}$$

(7)

The best value of control parameters which meet the requirements of coupling equations can be found by system coordination. In this study, this process is stated as an optimization problem

$$\begin{array}{c}\mathrm{Find}\mathbf{\lambda }=({\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2,\dots ,{\mathbf{\lambda }}_n)\hfill \\ \mathrm{satisfying}{\mathbf{Y}}_{i•}\left(\mathbf{\lambda }\right)={\mathbf{Y}}_{i•}({\mathbf{D}}_i\left(\mathbf{\lambda }\right),{\mathbf{Y}}_{•i}\left(\mathbf{\lambda }\right))\hfill \end{array}$$

(8)

Based on the Lagrange duality theorem, the optimization in equation (3) will be equivalent to the Lagrange dual optimization in equation (9), if there is a saddlepoint for equation (4)

$$\begin{array}{c}\underset{{\mathbf{D}}_i,{\mathbf{Y}}_{•i},{\mathbf{Y}}_{i•}}{min}f=\sum _{i=1}^n{f}_i({\mathbf{D}}_i,{\mathbf{Y}}_{•i},{\mathbf{Y}}_{i•})=\underset{\mathbf{\lambda }}{max}\varphi \hfill \\ =\sum _{i=1}^n{L}_i({\mathbf{D}}_i\left(\mathbf{\lambda }\right),{\mathbf{Y}}_{i•}({\mathbf{D}}_i\left(\mathbf{\lambda }\right),{\mathbf{Y}}_{•i}\left(\mathbf{\lambda }\right)),{\mathbf{Y}}_{•i}\left(\mathbf{\lambda }\right),\mathbf{\lambda })\hfill \end{array}$$

(9)

Note that the derivative of $\varphi (\mathbf{\lambda })$ with respect to $\mathbf{\lambda }$ is

$$\begin{array}{c}\varphi \prime \left(\mathbf{\lambda }\right)={\mathbf{Y}}_{•i}\left({\mathbf{Y}}_{i•}\left(\mathbf{\lambda }\right)\right)-{\mathbf{Y}}_{•i}\left(\mathbf{\lambda }\right)\hfill \\ \mathrm{where}{\mathbf{Y}}_{i•}\left(\mathbf{\lambda }\right)={\mathbf{Y}}_{i•}({\mathbf{Y}}_{•i}\left(\mathbf{\lambda }\right),{\mathbf{D}}_i\left(\mathbf{\lambda }\right))\hfill \end{array}$$

(10)

If $\varphi \prime (\mathbf{\lambda })=0$ , equation (10) will be the same as equation (2) and coupling equations will be satisfied. So, the task of the system coordination shown in equation (8) can be transformed into a problem for finding $\mathbf{\lambda }$ , to satisfy $\varphi \prime (\mathbf{\lambda })={\mathbf{Y}}_{•i}({\mathbf{Y}}_{i•}(\mathbf{\lambda }))-{\mathbf{Y}}_{•i}(\mathbf{\lambda })=0$ where ${\mathbf{Y}}_{i•}({\mathbf{D}}_i(\mathbf{\lambda }),{\mathbf{Y}}_{•i}(\mathbf{\lambda }))={\mathbf{Y}}_{i•}(\mathbf{\lambda })$ . The control parameters $\mathbf{\lambda }$ can be updated by iterative formula

$$\begin{array}{c}{\mathbf{\lambda }}_i^{k+1}={\mathbf{\lambda }}_i^k+{\alpha \frac{\partial \varphi \left(\mathbf{\lambda }\right)}{\partial {\mathbf{\lambda }}_i}|}_{\mathbf{\lambda }={\mathbf{\lambda }}^k}i=1,2,\dots ,n\hfill \\ ={\mathbf{\lambda }}_i^k+\alpha [{\mathbf{Y}}_{•i}\left({\mathbf{Y}}_{i•}({\mathbf{D}}_i^k\left({\mathbf{\lambda }}^k\right),{\mathbf{Y}}_{•i}^k\left({\mathbf{\lambda }}^k\right))\right)-{\mathbf{Y}}_{•i}^k\left({\mathbf{\lambda }}^k\right)]\hfill \end{array}$$

(11)

where $\alpha$ is the iterative coefficient and its value lies between 0 and 1; k presents the kth iterative. The two-level optimization structure of CRCA is shown in Figure 5. At system level, the system coordinator maintains the interdisciplinary consistency and updates the control parameters using equation (11). This process treats all discipline solutions ${\mathbf{D}}_i^k,{\mathbf{Y}}_{•i}^k$ as design parameters. After that, the updated control parameters ${\mathbf{\lambda }}_i^k$ are assigned to each discipline. The discipline objectives in equation (7) are optimized at sub-system level. This process treats control parameters ${\mathbf{\lambda }}_i^k$ as design parameters. The procedure of CRCA includes six steps, which are described below and also plotted in Figure 6:

• Step 1. Set initial values for control parameters ${\mathbf{\lambda }}_i^0$ for the iteration $k=1$ .

• Step 2. Assign control parameters ${\mathbf{\lambda }}_i^k$ to the disciplines by the system coordinator. Solve the disciplines optimization problems in equation (7) with the fixed ${\mathbf{\lambda }}_i^k$ and obtain the solutions ${\mathbf{D}}_i^k({\mathbf{\lambda }}^k)$ and ${\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)$ . Then, send them to the coordinator.

• Step 3. Calculate ${\mathbf{Y}}_{•i}({\mathbf{Y}}_{i•}({\mathbf{D}}_i^k({\mathbf{\lambda }}^k),{\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)))-{\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)$ , $i=1,2,\dots ,n$ and obtain the value of $|{\mathbf{Y}}_{•i}({\mathbf{Y}}_{i•}({\mathbf{D}}_i^k({\mathbf{\lambda }}^k),{\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)))-{\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)|$ .

• Step 4. Check the convergence. If objective does not change drastically and $|{\mathbf{Y}}_{•i}({\mathbf{Y}}_{i•}({\mathbf{D}}_i^k({\mathbf{\lambda }}^k),{\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)))-{\mathbf{Y}}_{•i}^k({\mathbf{\lambda }}^k)|\le \epsilon$ where $\epsilon$ is an arbitrary small positive constant, go to Step 6. Or, go to Step 5.

• Step 5. Obtain the new control parameters ${\mathbf{\lambda }}_i^{k+1}$ using the iterative formula as shown in equation (11) and then turn to Step 2 by setting $k=k+1$ .

• Step 6. End the procedure. Obtain the optimum solutions ${\hat{\mathbf{D}}}_i^k({\mathbf{\lambda }}^k)$ , ${\hat{\mathbf{Y}}}_{•i}^k({\mathbf{\lambda }}^k)$ , ${\hat{\mathbf{Y}}}_{i•}^k({\mathbf{\lambda }}^k)$ , ${\hat{f}}_i$ , and $\hat{\mathbf{\lambda }}$ which can satisfy coupling equations in equation (2).

Figure 5.

The two-level optimization structure of the CRCA strategy.

Figure 6.

The flowchart of CRCA.

Test example

The speed reducer design example is derived from the National Aeronautics and Space Administration (NASA) MDO evaluation examples which are shown in Figure 7. ³³ In this study, it is an MDO problem consisting of the bearing-shaft group 1 (discipline 1) and the bearing-shaft group 2 (discipline 2). The solutions obtained by CRCA are compared with the solutions obtained by CO and MDF, respectively. All optimization processes are conducted with MATLAB^™ software.

Figure 7.

The speed reducer design problem.

Table 1 shows the details of the design information. In this study, minimizing the speed reducer volume is chosen as the objective of this MDO problem. Correspondingly, minimizing the volume of the bearing-shaft group 1 and the bearing-shaft group 2 is chosen as the discipline objective. The original problem is modified into an MDO problem as shown in Figure 8. Detailed statements are provided in equations (12)–(14).

Table 1.

The details of design information.

Description	Range	Description	Range
The width of gear face $x_1$	$2.6\le x_1\le 3.6$	The distance between bearings 2 $x_5$	$7.3\le x_5\le 8.3$
The teeth module $x_2$	$0.3\le x_2\le 1.0$	The diameter of shaft 1 $x_6$	$2.9\le x_6\le 3.9$
The number of teeth of pinion $x_3$	$17\le x_3\le 28$	The diameter of shaft 2 $x_7$	$5\le x_7\le 5.5$
The distance between bearings 1 $x_4$	$7.3\le x_4\le 8.3$

Figure 8.

The MDO problem of speed reducer design.

1. Discipline 1 optimization problem

   $$\begin{array}{c}\mathrm{Find}{x}_1,x_2,x_3,x_6^{\left(1\right)},x_7^{\left(1\right)}\hfill \\ \min f_1=0.7854x_1{x}_2^2(3.333x_3^2+14.933x_3-43.0934)-1.508x_1({\left(x_6^{\left(1\right)}\right)}^2+{\left(x_7^{\left(1\right)}\right)}^2)\hfill \\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.g_1=27/\left(x_1{x}_2^2{x}_3\right)-1\le 0,\hfill & g_2=397.5/(x_1{x}_2^2{x}_3^2)-1\le 0\hfill \\ g_3=1.93x_4^3/(x_2{x}_3{\left(x_6^{\left(1\right)}\right)}^4)-1\le 0,\hfill & g_4=1.93x_5^3/(x_2{x}_3{\left(x_7^{\left(1\right)}\right)}^4)-1\le 0\hfill \\ g_5=A_1/B_1-1100\le 0,\hfill & g_6=A_2/B_2-850\le 0\hfill \\ g_7=x_2{x}_3-40\le 0,\hfill & g_8=x_1/x_2-12\le 0\hfill \\ g_9=-x_1/x_2+5\le 0,\hfill & g_{10}=(1.5x_6^{\left(1\right)}+1.9)/x_4-1\le 0\hfill \\ g_{11}=(1.1x_7^{\left(1\right)}+1.9)/x_5-1\le 0\hfill \end{array}\hfill \\ \begin{array}{cccc}2.6\le x_1\le 3.6,\hfill & 0.3\le x_2\le 1.0,\hfill & 17\le x_3\le 28,\hfill & 2.9\le x_6^{\left(1\right)}\le 3.9\hfill \\ \multicolumn{1}{c}{5\le x_7^{\left(1\right)}\le 5.5,}& x_6^{\left(1\right)}-x_6^{\left(2\right)}=0,\hfill & x_7^{\left(1\right)}-x_7^{\left(2\right)}=0\hfill \end{array}\hfill \end{array}$$

   (12)

2. Discipline 2 optimization problem

   $$\begin{array}{c}\mathrm{Find}{x}_4,x_5,x_6^{\left(2\right)},x_7^{\left(2\right)}\hfill \\ \min f_2=7.477({\left(x_6^{\left(2\right)}\right)}^3+{\left(x_7^{\left(2\right)}\right)}^3)+0.7854(x_4{\left(x_6^{\left(2\right)}\right)}^2+x_5{\left(x_7^{\left(2\right)}\right)}^2)\hfill \\ \begin{array}{cc}\mathrm{s}.\mathrm{t}.g_3=1.93x_4^3/(x_2{x}_3{\left(x_6^{\left(2\right)}\right)}^4)-1\le 0,\hfill & g_4=1.93x_5^3/(x_2{x}_3{\left(x_7^{\left(2\right)}\right)}^4)-1\le 0\hfill \\ g_5=A_1/B_1-1100\le 0,\hfill & g_6=A_2/B_2-850\le 0\hfill \\ g_9=-x_1/x_2+5\le 0,\hfill & g_{10}=(1.5x_6^{\left(2\right)}+1.9)/x_4-1\le 0\hfill \\ g_{11}=(1.1x_7^{\left(2\right)}+1.9)/x_5-1\le 0,\hfill & 7.3\le x_4,x_5\le 8.3\hfill \\ 2.9\le x_6^{\left(2\right)}\le 3.9,\hfill & 5\le x_7^{\left(2\right)}\le 5.5\hfill \\ x_6^{\left(1\right)}-x_6^{\left(2\right)}=0,\hfill & x_7^{\left(1\right)}-x_7^{\left(2\right)}=0\hfill \end{array}\hfill \end{array}$$

   (13)

where

$$\begin{array}{c}{A}_1={[{\left(\frac{745x_4}{x_2{x}_3}\right)}^2+16.9\times {10}^6]}^{0.5},B_1=0.1x_6^3\hfill \\ A_2={[{\left(\frac{745x_5}{x_2{x}_3}\right)}^2+157.5\times {10}^6]}^{0.5},B_2=0.1x_7^3\hfill \end{array}$$

(14)

Herein, sharing variables $x_6$ and $x_7$ are considered as discipline inputs, which can be denoted as $x_6^{(1)},x_7^{(1)}$ (input design variables for discipline 1) and $x_6^{(2)},x_7^{(2)}$ (input design variables for discipline 2), respectively. To maintain the consistency of sharing design variables in both disciplines, we assign $x_6^{(1)},x_7^{(1)}$ and $x_6^{(2)},x_7^{(2)}$ to discipline 2 and discipline 1, respectively. Then, extra equality constraints $x_6^{(1)}-x_6^{(2)}=0$ and $x_7^{(1)}-x_7^{(2)}=0$ are added in each discipline optimization problem, as shown in equations (12) and (13).

More specially, for the CRCA approach, this MDO problem is modified into a coordination problem and two discipline design problems. Using CRCA, we change equality constraints $x_6^{(1)}-x_6^{(2)}=0$ and $x_7^{(1)}-x_7^{(2)}=0$ into $x_6^{(1)}=x_6^{(2)}$ and $x_7^{(1)}=x_7^{(2)}$ . Then, in discipline 1, $x_6^{(2)}$ and $x_7^{(2)}$ can be considered as the function of $x_6^{(1)}$ and $x_7^{(1)}$ . Thus, they can be denoted as $x_6^{(2)}=x_6^{(2)}(x_6^{(1)})=x_6^{(1)}$ and $x_7^{(2)}=x_7^{(2)}(x_7^{(1)})=x_7^{(1)}$ , and vice versa in discipline 2. As special coupling design variables, sharing design variables in two disciplines can be coordinated by control parameters. Furthermore, the discipline objective functions can be changed into discipline Lagrange functions using equation (6). The framework of the problem is shown in Figure 9. Detailed statements of CRCA are given in equations (15)–(17).

• 1. The system coordination optimization problem

  $$\begin{array}{c}\mathrm{Find}{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\hfill \\ \max L=0.7854x_1{x}_2^2(3.333x_3^2+14.933x_3-43.0934)-1.508x_1({\left(x_6^{\left(1\right)}\right)}^2+{\left(x_7^{\left(1\right)}\right)}^2)\hfill \\ \begin{array}{c}\mathbf{+}7.477({\left(x_6^{\left(2\right)}\right)}^3+{\left(x_7^{\left(2\right)}\right)}^3)+0.7854(x_4{\left(x_6^{\left(2\right)}\right)}^2+x_5{\left(x_7^{\left(2\right)}\right)}^2)+{\lambda }_1(x_6^{\left(1\right)}-x_6^{\left(2\right)})\hfill \\ \multicolumn{1}{c}{+{\lambda }_2(x_7^{\left(1\right)}-x_7^{\left(2\right)})+{\lambda }_3(x_6^{\left(2\right)}-x_6^{\left(1\right)})+{\lambda }_4(x_7^{\left(2\right)}-x_7^{\left(1\right)})}\end{array}\hfill \end{array}$$

  (15)

• 2a. Optimization problem for discipline 1

  $$\begin{array}{c}\mathrm{Find}{x}_1,x_2,x_3,x_6^{\left(1\right)},x_7^{\left(1\right)}\hfill \\ \min L_1\left(\begin{array}{c}{x}_1,x_2,x_3,\hfill \\ \multicolumn{1}{c}{x_6^{\left(1\right)},x_7^{\left(1\right)}}\end{array}\right)=0.7854x_1{x}_2^2(3.333x_3^2+14.933x_3-43.0934)\hfill \\ -1.508x_1({\left(x_6^{\left(1\right)}\right)}^2+{\left(x_7^{\left(1\right)}\right)}^2)\mathbf{+}{\lambda }_1{x}_6^{\left(1\right)}+{\lambda }_2{x}_7^{\left(1\right)}-{\lambda }_3{x}_6^{\left(2\right)}-{\lambda }_4{x}_7^{\left(2\right)}\hfill \\ =0.7854x_1{x}_2^2(3.333x_3^2+14.933x_3-43.0934)\hfill \\ -1.508x_1({\left(x_6^{\left(1\right)}\right)}^2+{\left(x_7^{\left(1\right)}\right)}^2)\mathbf{+}{\lambda }_1{x}_6^{\left(1\right)}+{\lambda }_2{x}_7^{\left(1\right)}-{\lambda }_3{x}_6^{\left(1\right)}-{\lambda }_4{x}_7^{\left(1\right)}\hfill \\ \mathrm{s}.\mathrm{t}.g_1~g_{11},2.6\le x_1\le 3.6,0.3\le x_2\le 1.0,17\le x_3\le 28,2.9\le x_6^{\left(1\right)},x_6^{\left(2\right)}\le 3.9,\hfill \\ 5\le x_7^{\left(1\right)},x_7^{\left(2\right)}\le 5.5,x_6^{\left(2\right)}=x_6^{\left(2\right)}\left(x_6^{\left(1\right)}\right)=x_6^{\left(1\right)},x_7^{\left(2\right)}=x_7^{\left(2\right)}\left(x_7^{\left(1\right)}\right)=x_7^{\left(1\right)}\hfill \end{array}$$

  (16)

• 2b. Optimization problem for discipline 2

  $$\begin{array}{c}\mathrm{Find}{x}_4,x_5,x_6^{\left(2\right)},x_7^{\left(2\right)}\hfill \\ \min L_2\left(\begin{array}{c}{x}_4,x_5,\hfill \\ \multicolumn{1}{c}{x_6^{\left(2\right)},x_7^{\left(2\right)}}\end{array}\right)=7.477({\left(x_6^{\left(2\right)}\right)}^3+{\left(x_7^{\left(2\right)}\right)}^3)+0.7854(x_4{\left(x_6^{\left(2\right)}\right)}^2+x_5{\left(x_7^{\left(2\right)}\right)}^2)\hfill \\ +{\lambda }_3{x}_6^{\left(2\right)}+{\lambda }_4{x}_7^{\left(2\right)}-{\lambda }_1{x}_6^{\left(1\right)}-{\lambda }_2{x}_7^{\left(1\right)}\hfill \\ =7.477({\left(x_6^{\left(2\right)}\right)}^3+{\left(x_7^{\left(2\right)}\right)}^3)+0.7854(x_4{\left(x_6^{\left(2\right)}\right)}^2+x_5{\left(x_7^{\left(2\right)}\right)}^2)\hfill \\ +{\lambda }_3{x}_6^{\left(2\right)}+{\lambda }_4{x}_7^{\left(2\right)}-{\lambda }_1{x}_6^{\left(2\right)}-{\lambda }_2{x}_7^{\left(2\right)}\hfill \\ \begin{array}{ccc}\mathrm{s}.\mathrm{t}.g_3~g_6,g_9~g_{11},\hfill & 7.3\le x_4,x_5\le 8.3,\hfill & 2.9\le x_6^{\left(1\right)},x_6^{\left(2\right)}\le 3.9,\hfill \\ 5\le x_7^{\left(1\right)},x_7^{\left(2\right)}\le 5.5,\hfill & x_6^{\left(1\right)}=x_6^{\left(1\right)}\left(x_6^{\left(2\right)}\right)=x_6^{\left(2\right)},\hfill & x_7^{\left(1\right)}=x_7^{\left(1\right)}\left(x_7^{\left(2\right)}\right)=x_7^{\left(2\right)}\hfill \end{array}\hfill \end{array}$$

  (17)

where $L$ denotes the optimization objective; $L_1$ and $L_2$ denote the discipline objectives; ${\lambda }_1,{\lambda }_2,{\lambda }_3,\mathrm{and}{\lambda }_4$ are control parameters; $x_1,x_2,x_3,x_6^{(1)},\mathrm{and}{x}_7^{(1)}$ are inputs of bearing-shaft group 1; and $x_4,x_5,x_6^{(2)},\mathrm{and}{x}_7^{(2)}$ are inputs of bearing-shaft group 2.

Figure 9.

The application of CRCA approach in the test example.

Here, the MDO problem is solved at two different initial points: point 1 (2.65, 0.63, 18, 6.80, 6.400, 3.00, 5.099) and point 2 (3.50, 0.70, 17, 7.30, 7.715, 3.35, 5.287), respectively.

Initial control parameters are ${\mathbf{\lambda }}_1$ and ${\mathbf{\lambda }}_2$

$${\mathbf{\lambda }}_1=({\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2,{\mathbf{\lambda }}_3,{\mathbf{\lambda }}_4)=(0,0,0,0)$$

$${\mathbf{\lambda }}_2=({\mathbf{\lambda }}_1,{\mathbf{\lambda }}_2,{\mathbf{\lambda }}_3,{\mathbf{\lambda }}_4)=(1,-1,1,-1)$$

The results are compared with solutions obtained by different MDO methods, which are shown in Tables 2 and 3. $n_s$ is the iteration number at the system level, and $n_1$ and $n_2$ are the iteration numbers of discipline problem 1 and 2, respectively. The corresponding discipline optimization problems at the sub-system level in the both methods have the different objectives. The number of system coordination iterations and numbers of the discipline iterations of the CRCA method are much less than those of the CO method. Moreover, the corresponding discipline optimization problems in both CRCA and CO have the same design variables. It should be noted that not only the discipline optimization objectives but also the design parameters in both methods are not the same. For the CRCA method, in each discipline optimization problem, the discipline design parameters are the control parameters. For the CO method, in each discipline optimization problem, the discipline design parameters are the design variables from system coordination problem. Hence, the discipline parameters number in CRCA can be much less than that of the CO method. With the increasing amount of design input information, this difference between the CRCA method and CO method will be more significant.

Table 2.

Design solutions from different MDO methods.

	Point 1				Point 2
	CO	MDF	CRCA $({\mathbf{\lambda }}_1)$	CRCA $({\mathbf{\lambda }}_2)$	CO	MDF	CRCA $({\mathbf{\lambda }}_1)$	CRCA $({\mathbf{\lambda }}_2)$
$x_1$	3.478	3.600	3.584	3.587	3.495	3.600	3.600	3.578
$x_2$	0.630	0.664	0.665	0.665	0.664	0.664	0.664	0.666
$x_3$	18	17	17	17	17	17	17	17
$x_4$	7.307	7.300	7.300	7.300	7.300	7.300	7.300	7.300
$x_5$	7.781	7.715	7.715	7.715	7.715	7.715	7.715	7.715
$x_6$	3.361	3.351	3.350	3.350	3.347	3.351	3.375	3.353
			3.354	3.354			3.350	3.351
$x_7$	5.305	5.287	5.286	5.286	5.279	5.287	5.286	5.286
			5.286	5.286			5.286	5.286
f	2852	2874	2875	2874	2745	2872	2882	2876

MDO: multidisciplinary design optimization; CO: collaborative optimization; MDF: multidisciplinary feasibility; CRCA: coupling relationship coordination algorithm.

Table 3.

Control parameters and the iteration number.

	Point 1			Point 2
	CO	CRCA $({\mathbf{\lambda }}_1)$	CRCA $({\mathbf{\lambda }}_2)$	CO	CRCA $({\mathbf{\lambda }}_1)$	CRCA $({\mathbf{\lambda }}_2)$
${\mathbf{\lambda }}_1$	−	15.02	15.51	−	14.71	14.69
${\lambda }_2$	−	−249.99	−247.28	−	−244.02	−250.08
${\lambda }_3$	−	−249.74	−250.00	−	−248.82	−253.65
${\lambda }_4$	−	−247.99	−247.30	−	−243.73	−246.07
$n_s$	98	22	28	55	15	20
$n_1$	2362	809	783	1117	774	837
$n_2$	2412	419	427	1029	236	406

Conclusion

In this study, a control and coordination method of large-scale systems is introduced into MDO and the corresponding formulation of CRCA is proposed. The application of CRCA into MDO has several advantages including: (1) enjoying less transferred variables and parameters between coupling disciplines; (2) removing the MDA of complex engineering systems; (3) employing professional experts and discipline software for different discipline optimization problems; and (4) providing a parallelized analysis and design strategy, which is more suited for modern discipline distribution. An example is utilized to illustrate the application of CRCA in MDO.